Physics Revision: Vectors and Scalars
... 4. A resultant is one vector, which has the same effect on a body as the two or more vectors that are actually acting on that body. It starts at the beginning of the first vector and ends at the end of the last one. 5. An equilibrant is one vector, which cancels out the effect that the two or more v ...
... 4. A resultant is one vector, which has the same effect on a body as the two or more vectors that are actually acting on that body. It starts at the beginning of the first vector and ends at the end of the last one. 5. An equilibrant is one vector, which cancels out the effect that the two or more v ...
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... A space X is disconnected if it has disjoint open subsets U and V with U ∩ V = ∅ and U ∪ V = X. We say that points x, y ∈ X are in the same component, written x ∼ y, if there exists a connected C ⊂ X with x, y ∈ C. The equivalence classes are called components, and X is the disjoint union of its com ...
... A space X is disconnected if it has disjoint open subsets U and V with U ∩ V = ∅ and U ∪ V = X. We say that points x, y ∈ X are in the same component, written x ∼ y, if there exists a connected C ⊂ X with x, y ∈ C. The equivalence classes are called components, and X is the disjoint union of its com ...
Equations for the vector potential and the magnetic multipole
... where m = Ia is the magnetic dipole moment of the current loop. I am going to skip over the higher multipoles in these notes. Instead, let me consider replacing a single wire loop with a circuit of several connected wires. In this case, we may use the Kirchhoff Law to express the whole circuit as se ...
... where m = Ia is the magnetic dipole moment of the current loop. I am going to skip over the higher multipoles in these notes. Instead, let me consider replacing a single wire loop with a circuit of several connected wires. In this case, we may use the Kirchhoff Law to express the whole circuit as se ...
Continuous operators on Hilbert spaces 1. Boundedness, continuity
... • Appendix: topologies on finite-dimensional spaces Among all linear operators on Hilbert spaces, the compact ones (defined below) are the simplest, and most closely imitate finite-dimensional operator theory. In addition, compact operators are important in practice. We prove a spectral theorem for ...
... • Appendix: topologies on finite-dimensional spaces Among all linear operators on Hilbert spaces, the compact ones (defined below) are the simplest, and most closely imitate finite-dimensional operator theory. In addition, compact operators are important in practice. We prove a spectral theorem for ...
CHAP03 Vectors and Matrices in 3 Dimensions
... We defined the determinant of a 2 × 2 matrix as the signed area of a certain parallelogram. We shall define the determinant of a 3 × 3 matrix as the signed volume of a certain parallelepiped. Of course these definitions can only work if the components of the matrices are real numbers. In a later cha ...
... We defined the determinant of a 2 × 2 matrix as the signed area of a certain parallelogram. We shall define the determinant of a 3 × 3 matrix as the signed volume of a certain parallelepiped. Of course these definitions can only work if the components of the matrices are real numbers. In a later cha ...
Math for Programmers
... • A transformation T:VW is a function that maps elements from vector space V to W • The function f(x, y) = x2 + 2y is a transformation because it maps R2 into R ...
... • A transformation T:VW is a function that maps elements from vector space V to W • The function f(x, y) = x2 + 2y is a transformation because it maps R2 into R ...
n-Dimensional Euclidean Space and Matrices
... Definition of n space. As was learned in Math 1b, a point in Euclidean three space can be thought of in any of three ways: (i) as the set of triples (x, y, z) where x, y, and z are real numbers; (ii) as the set of points in space; (iii) as the set of directed line segments in space, based at the ori ...
... Definition of n space. As was learned in Math 1b, a point in Euclidean three space can be thought of in any of three ways: (i) as the set of triples (x, y, z) where x, y, and z are real numbers; (ii) as the set of points in space; (iii) as the set of directed line segments in space, based at the ori ...
A NICE PROOF OF FARKAS LEMMA 1. Introduction Let - IME-USP
... theorem valid for vector spaces of dimension less than dim(V ). We will show that the theorem holds for V . This will be done by induction on the number k of vectors on the list v1 , . . . , vk . If k = 1, we consider two possibilities: if v1 and x are linearly independent, there exists a linear fun ...
... theorem valid for vector spaces of dimension less than dim(V ). We will show that the theorem holds for V . This will be done by induction on the number k of vectors on the list v1 , . . . , vk . If k = 1, we consider two possibilities: if v1 and x are linearly independent, there exists a linear fun ...
Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied (""scaled"") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below. Euclidean vectors are an example of a vector space. They represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are regarded as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows.Vector spaces are the subject of linear algebra and are well understood from this point of view since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. A vector space may be endowed with additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide whether a sequence of vectors converges to a given vector. This is accomplished by considering vector spaces with additional structure, mostly spaces endowed with a suitable topology, thus allowing the consideration of proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory.Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs.Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear-algebraic notion to deal with systems of linear equations; offer a framework for Fourier expansion, which is employed in image compression routines; or provide an environment that can be used for solution techniques for partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra.